A unimodular matrix $M$ is a square integer matrix having determinant $+1$ or $−1$.
A totally unimodular matrix (TU matrix) is a matrix for which every square non-singular submatrix is unimodular. A totally unimodular matrix need not be square itself. Obviously, any totally unimodular matrix has only $0$, $+1$ or $−1$ entries.

Now suppose a $n\times n$ non-singular matrix $A$ is totally unimodular. Can we prove that
$A^{-1}$ is also totally unimodular? Or if it is not correct, can we have a counterexample?

@Gerhard: The inverse of the matrix that you've mentioned is also TU...so I don't understand your comment???
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SuvritApr 19 '13 at 22:25

S. Sra, I am suggesting that I don't know what group structure you are placing on the set of matrices. For the two notions of multiplication I considered, the matrix I gave does not help form a group. Gerhard "How Do You Multiply Them?" Paseman, 2013.04.19
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Gerhard PasemanApr 19 '13 at 22:52

1 Answer
1

The answer is yes, because if $B=A^{-1}$, then we have an equality between minors:
$$B(I,J)=\pm\frac{A(J^c,I^c)}{\det A},$$
for every subsets $I,J\subset[[1,n]]$ of same cardinals. This formula generalizes that giving the entries of $A^{-1}$ in terms of minors of $A$. The $\pm$ sign is not essential to prove the stability of the TU class under inversion.